6.1 Analysis of frequency selective surfaces

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1 6.1 Aalysis of frequecy selective surfaces Basic theory I this paragraph, reflectio coefficiet ad trasmissio coefficiet are computed for a ifiite periodic frequecy selective surface. The attetio is tured to the surface cosistig of rectagular elemets (metallic, slot). First, the surface is assumed to be fabricated from such dielectrics, which properties are idetical with surroudig. At the ed, a brief ote o the aalysis of real substrates is provided. Numeric aalysis of frequecy selective surfaces There are two approaches to the aalysis of frequecy selective surfaces. The first oe is based o the method of iduced electromotoric forces ad eables to aalyze surfaces of both the fiite extet ad the ifiite oe (i most cases, the method is applied to ifiite structures). Aalysis by the method of iduced electromotoric forces is ot described here; readers iterested ca fid more iformatio i [17]. Spectral domai momet method is the secod way of aalyzig frequecy selective surfaces [18]. We cocetrate to this method oly. Fig. 6.1A.1 Coordiate system for the aalysis of a frequecy selective surface (left). A elemetary cell of the dimesios a, b (right). Metallic elemets We assume a periodic surface as depicted i fig. 6.1A.1. The surface is illumiated by a plae wave. Projectio of wave vector k to the plae of the surface is deoted by k xy. Electric filed itesity of icidet wave ca be expressed the as follows: E I = E 0 exp[+ j(α 0 x + β 0 y)]. ( 6.1A.1 ) Here, α 0, β 0 are egative projectios of wave vector of icidet wave to the directios of coordiate axes x ad y (the expoetial term exp(+jkr) is assumed for icreasig phase i the directio of propagatio). Projectios of wave vector i the described coordiate system are give as follows: α 0 = k si(ϑ)cos(φ), β 0 = k si(ϑ)si(φ). The symbol E 0 deotes electric-field itesity vector i the origi of the coordiate system ad k is free-space wave umber. I geeral, the icidet wave is of both the parallel polarizatio ad the perpedicular oe. If field itesity of the icidet wave is expressed i spherical coordiate I system (fig. 6.1A.1), the compoets E ϑ ad I Eφ represet directly the parallel compoet ad the perpedicular oe. I If frequecy selective surface is goig to be aalyzed, compoets of electric field itesity i the plae xy have to be kow. We ca evaluate them re-computig E ϑ I ad E φ as follows: EI x si(ϑ)cos(φ) si(φ) EI ϑ = EI. ( 6.1A. ) y si(ϑ)si(φ) cos(φ) EI φ Due to the periodicity of the surface, all the ecessary quatities (electric field itesity, curret desity) i the plae xy ca be expressed usig Fourier series. Usig Fourier expasio, desity of electric curret J [A/m] o metallic elemet is expressed: M N J = J(αm, β )exp[+ j(α m x + β y)], ( 6.1A.3 ) m = M = N where α = α 0 + π a m, β = β 0 + π b ( 6.1A.4 ) are spatial frequecies. Those spatial frequecies are a spatial aalogy to the temporal agular frequecy ω, which are used for temporal aalysis of sigals. The oly differece is hidde i the fact that time period T correspods to spatial periods a ad b. Coefficiets of Fourier series ca be obtaied by itegratig curret desities over the whole surface of a cell (iverse Fourier trasform) J(α m, β ) = 1 ab J(x, y)exp[ j(α m x + β y)]. ( 6.1A.5 ) plocha bu ň ky I aalogy to curret desity, Fourier series ca express arbitrary periodic quatity. We do that i the ext part, where a plae wave excitatio of a selective surface is expressed i the spectral domai (i.e., i the domai of spatial frequecies). Determiatio of reflective properties of the selective surface is the aim of the computatio.

2 Problem formulatio Assume a plae wave fallig to a frequecy selective surface uder agles (ϑ, φ). This wave excites currets i metallic elemets, which form a scattered field E S. Whereas the magitude (ot the phase) of the itesity of icidet wave E I is costat over the whole elemetary cell, the magitude of the itesity of the scattered wave E S is differet i the differet poits of the cell. Relatio betwee E S ad E I chages accordig to the surface impedace i a give poit. For the surface of coductive elemets, we get E I + E S = J γ. ( 6.1A.6 ) Here, γ [S.m -1 ] is electric coductivity of elemets. Assumig the electric coductivity of metallic elemets to be perfect, the right-had side of (6.1A.6) approaches zero, ad we get E I + E S = 0. ( 6.1A.7 ) Itesity of scattered electric field E S i a poit determied by the positio vector r ca be evaluated from vector potetial A E S (r) = jωµ { A + 1 k [ A(r)] }. ( 6.1A.8 ) Fig. 6.1A. Frequecy selective surface (reflected ad trasmitted wave) As a source of vector potetial A, currets J iduced by icidet wave to metallic elemets of selective surface ca be cosidered. Cotributio of curret i r' to the potetial i r is described by Gree fuctio G. For vector potetial, we get I free space, Gree fuctio G is of the form Combiig relatios (6.1A.9), (6.1A.8) ad (6.1A.7), itegral equatio for space-domai electric field is obtaied: A(r) = G(r r )J(r )ds. ( 6.1A.9 ) zdro je G = exp( jk r r ) 4π r r. ( 6.1A.10 ) jωµ G(r r ) J(r )ds + 1 { zdro je k [ G(r r ) J(r )ds zdro je ]} = EI. ( 6.1A.11 ) Eq. (6.1A.11) is solved i the domai of spatial frequecies. Eq. (6.1A.11) is mapped to the spectral domai withi two steps. First, eq. (6.1A.8) is mapped by its rewritig to the compoet form ad by expressig vector potetial by Fourier series. That way, we get [19] ES x (αm, β 1 α m ) k = jωµ ES z (αm, β ) αm β k I the secod step, the relatio is mapped to the domai of spatial frequecies (6.1A.9) [19] Evaluatig Fourier trasform of the fuctio (6.1A.10), we get [19] Gree fuctio i spectral domai where the square-root of egative imagiary part is assumed oly. αm β k A x(αm, β ) 1 β. ( 6.1A.1 ) A y(αm, β ) k A(α m, β ) = G(α m, β ) J(α m, β ). ( 6.1A.13 ) G(α m, β ) = j k αm β Combiig (6.1A.13), (6.1A.14) ad (6.1A.1), the fial relatio for electric field i spectral domai is obtaied k α m k 1 ωε α m β m, α m β k α m β α m β k α m β k β k α m β, ( 6.1A.14 ) J x (αm, β ) EI x (x, y) exp[ j(α m x + β y)] =. ( 6.1A.15 ) J y(αm, β ) EI y(x, y) Problem solutio Solvig the problem, a ukow curret distributio i (6.1A.15) is approximated by the liear combiatio of properly chose basis fuctios ad ukow approximatio coefficiets. Such formal approximatio is substituted ito the solved eq. (6.1A.15). Sice the approximatio of the solutio does ot meet the iitial relatio exactly, we respect this fact by itroducig a residual fuctio (residuum). Lower the residuum is, more accurate the approximatio is. The residuum is miimized by Galerki method (residuum is sequetially multiplied by as may basis fuctios as may ukow approximatio coefficiets is computed; that way, N liear algebraic equatios for N ukow approximatio coefficiets is obtaied). There are two approaches for choosig basis fuctios. The first oe exploits basis fuctios, which are o-zero over the whole aalyzed area. Fuctios are elected to physically represet stadig waves of a curret o a elemet. The secod approach divides the aalyzed regio to sub-regios, where curret is approximated i terms of basis fuctios, which fuctioal value is o-zero over a

3 give sub-regio oly. This approach is advatageous i simple aalysis of selective surfaces cosistig of arbitrarily shaped elemets. Here, we cocetrate o the first group of basis fuctios oly. As basis fuctios, harmoic fuctios are used (first approach, the layer B), ad a combiatio of harmoic fuctios ad Chebyshev polyomials is exploited (secod approach, the layer B). A more detailed descriptio of the above-give basis fuctios for solvig (6.1A.15) is itroduced i the layer B. Here, obtaied results are preseted oly. Harmoic basis fuctios Assume a frequecy selective surface cosistig of rectagular perfectly coductive elemets of the dimesios a' ad b'. If curret distributio over a elemet is approximated i terms of harmoic basis fuctios, we obtai for the mode (1, 1) for the parallel polarizatio such curret distributio, which is depicted i fig. 6.1A.3a, ad for the perpedicular polarizatio, the distributio from fig. 6.1A.3b is obtaied a) b) Fig. 6.1A.3 Directioal field of curret desity of the mode (1,1) a) for parallel polarizatio b) for perpedicular oe. The cell is of dimesios a = b = 1mm, the metallic elemet is of dimesios a' = b' = 1mm; f=10ghz, ϑ = 1, φ = 45, d = 1.5mm, ε r = 3.7. Magitude of margial currets was icreased with respect to reality. Plae of icidece is depicted i red. Combiatio of harmoic fuctios ad Chebyshev polyomials I cotrast to purely harmoic basis fuctios, the fuctio cosie is replaced by Chebyshev polyomial, which is of ifiite magitude at the edges of the elemet. That way, strog currets flowig alog edges are correctly modeled. A equivalet circuit ca describe behavior of frequecy selective surfaces, which cosist of metallic elemets, where iductaces of elemets together with capacitaces amog eds of elemets create a serial resoat circuit (fig. 6.1A.4). Reflectio coefficiet r (see fig. 6.1A.4) is evaluated accordig to P T = P I 1 ρ. ( 6.1A.16 ) Here, P I is power of icidet wave ad P T deotes power of wave, which was trasmitted by the surface to the right-had half-space. Resistace 377 Ω models free space i frot of the surface (left) ad behid the surface (right). Fig. 6.1A.4 Equivalet circuit of selective surface i frequecy domai; surface cosist of metallic elemets (left); ϑ = φ = 0. Directivity factor i Smith chart (right).

4 a) b) Fig. 6.1A.5 Surface of rectagular elemets, tued for 10 GHz, parallel polarizatio a) ormal icidece ϑ = 0, φ = 90 b) uormal icidece ϑ = 45, φ = 90 Frequecy course of reflectio coefficiet of the surface for parallel polarizatio is depicted i fig. 6.1A.5. Cosiderig the surface without dielectrics, (b = 1 mm, a = 7.5 mm, b' = mm, a' = 1.5 mm), there is a break of the steepess of the reflectio coefficiet course aroud 14.3 GHz; this pheomeo is caused by the excitatio of the first parasitic mode (gratig lobe). Addig dielectrics ad modifyig dimesios (b = 17 mm, a = 7.5 mm, b' = mm, a' = 1.5 mm, height of the substrate d = 1.57 mm) causes a small decrease of the selectivity o oe had, but the stability of tuig for u-ormal icidece is icreased o the other had (see fig. 6.1A.5b). Further icrease of the stability of reflectio coefficiet with respect to the agle of icidece ca be reached by addig a upper dielectric layer. Thaks to the dielectrics, cell dimesios are smaller ad parasitic modes are excited at higher frequecies (17.6 GHz). I fig. 6.1A.5b, o sigificat effect of this mode ca be observed. Both the ormal icidece ad the u-ormal oe excite parasitic resoace at higher frequecies. At frequecies, which are higher tha the frequecy of the first parasitic mode, the course of reflectio coefficiet does ot express the total itesity of the reflected wave, but oly the itesity of the basic mode (0,0). Cosiderig perpedicular polarizatio of icidet wave (i cotrast to parallel oe), reflectio coefficiet is early idepedet o the agle of icidece (see fig. 6.1A.6). If a surface without dielectrics is cosidered, dimesios b = 1 mm, a = 7.5 mm, b' = mm, a' = 1.5 mm, are assumed., If dielectrics is cosidered, dimesios b = 17 mm, a = 7.5 mm, b' = mm, a' = 1.5 mm ad the height of the substrate d = 1.57 mm are assumed. Fig. 6.1A.6 Surface of rectagular elemets, tued for 10 GHz, uormal icidece, ϑ = 45, φ = 0, perpedicular polarizatio. Slot elemets I cotrast to the surfaces of metallic elemets, frequecy selective surfaces of slots exhibit a opposite depedecy of reflectio coefficiet, i.e. they behave as a bad-stop filter. If o dielectrics ad a sigle metallic layer are assumed, the slot surfaces are true complemets of metallic-elemet surfaces.

5 surface 1 surface Fig. 6.1A.7 Complemetary behavior of slot surfaces ad metallic-elemet oes. Reflectio coefficiet of metallic-elemet surface (1) equals to trasmissio coefficiet of slot surface (). Now, we tur our attetio to the derivatio of equatios for magetic itesity over the aperture of slot surface i spectral domai. Cosiderig the case without dielectrics ad with a sigle metallic layer, duality priciple ca be applied: i (6.1A.15), we exchage E with H, ε with µ, J with J M k α m k 1 ωµ α m β m, α m β k α m β α m β k α m β k β k α m β J M + x (αm, β ) H I x (x, y) exp[ j(α m x + β y)] =. ( 6.1A.17 ) J M + y (αm, β ) H I y (x, y) Comparig (6.1A.17) ad (6.1A.15), a certai icosistecy ca be revealed: coefficiet appeared at magetic curret desity ad excitig magetic itesity. Next to the duality priciple, the followig fact had to be cosidered: Field radiated by electric currets of the metallic-elemet surface is ot scattered by those elemets, whereas magetic currets i slots produce a field, which is scattered by the coductive plae [19]. I order to iclude the ifluece of the surroudig of the coductive plae, followig steps have to be doe: Origial electric currets are replaced by magetic oes (flowig o the surface of the slot). Magetic curret desity ca be computed applyig J M = z Eap, where E ap is electric field itesity i the aperture (similarly to electric curret, eve the magetic coductor of a real thickess is flow by the curret from oe side oly - from left oe). The slot is assumed to be filled i by perfect electric coductor (that way, a ifiite cotiuous groud plae is obtaied). This assumptio does ot chage the situatio ay way because at the boudary of metal ad air, the tagetial compoets of electric field itesity are related by Ekov t Et vzduch = JM + ( z), ad field itesity i metal Ekov t = 0 is zero. Mirror priciple is applied, which coverts the problem to free-space oe (fig. 6.1A.8 right). origial problem whe magetic curret itroduced extesio of groud plae ad cosiderig left-had half-space oly Fig. 6.1A.8 Explaatio of equivalece priciple applicatio of mirror priciple I aalogy to metallic elemets, dual formulatio of trasmissio coefficiet is possible τ. I fig. 6.1A.9, a example for ormal icidece of a wave to a frequecy selective surface with slots i vacuum (b = 1 mm, a = 7.5 mm, b' = mm, a' = 1.5 mm) ad o dielectric substrate (b = 14 mm, a = 7.5 mm, b' = 13.5 mm, a' = 1.5 mm ad the height of the substrate d = 1.57 mm) are give. Obviously, selectivity of trasmissio coefficiet is better for dielectrics (i case of metallic-elemet surface, the situatio was opposite). Next, ifluece of dielectrics is differet compared to metallic-elemet surface, ad therefore, slots are of differet dimesios compared to metallic-elemet surface with dielectrics. Dielectrics eve cause o-zero value of reflectio coefficiet o resoat frequecy. Eve here, parasitic modes appear. Coclusio

6 The method of the aalysis of frequecy selective surfaces, give i this chapter, does ot iclude the ifluece of dielectrics, but give examples do that [17]. Aother way of the descriptio of the ifluece of dielectrics ca be foud i [19]. Real applicatios use several sequetially lied selective surfaces. That way, selectivity ca be sigificatly improved. Desig of such surfaces is rather complicated. Frequecy selective surfaces ca be modeled i commercial programs based o fiite elemets (HFSS) or o momet method (IE3D). I both cases, oly a sigle elemet is aalyzed thaks to the periodicity. Desigig frequecy selective surfaces, physical basis of their behavior ad limitatios have to be kow. trasmissio reflectio Fig. 6.1A.9 Surface of rectagular slots, tued for 10 GHz, ormal icidece ϑ = 0 φ = 90, perpedicular polarizatio.